Suppose you had a dynamical system $\dot{\vec{x}} = \vec{f}(\vec{x})$. In theory, one could represent this as a directed graph where the vertices are fixed points of the dynamical system and the edges of the graph are the orbits between them.
I've read about book embeddings of graphs, but other than it being just a nice visual representation, does a book embedding tell you anything about the dynamical system?
Likewise, I've been reading about the adjacency matrix of a graph. It's a nice way to convert a graph into a matrix in which one knows a lot of algebra about, but does anything about this matrix give you insight about the dynamical system? (e.g. determinants, eigenvalues, etc.). I've seen papers that give bounds on the determinants of graphs, but it doesn't sound like that bound gives you any information about the dynamical system.
It seems interesting to be able to compare graphs and dynamical systems, but to me at least, it just seems like a graph is a nice visualization of dynamical system behavior, and I'm not sure what quantitative information you can extract from the graph. I'm not a graph theorist so I'm not familiar with this field, but I'd be quite interested in finding out quantitative relationships between a graph and dynamical systems.
This idea is not new. You may take a look at the following paper, for example http://epubs.siam.org/doi/abs/10.1137/080734935 .
There is also something you can do for chaotic system if you use Ulam method to approximate the transfer operator of the system. Then you can measure the force of chaoticity of the system in a sense, and approximate its invariant measure.